Mathematics Exam > Mathematics Questions > Which of the following(s) is/are correct?a)Th...
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Which of the following(s) is/are correct?
- a)The transpose of a symmetric matrix need not be summetric matrix.
- b)If A and B are symmetric matrix of same order, then AB + BA must be symmetric matrix.
- c)If A is symmetric matrix, then all positive integral powers of A are symmetric matrices.
- d)If A is any square matrix, then A + A'is always symmetric
Correct answer is option 'B,C,D'. Can you explain this answer?
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Which of the following(s) is/are correct?a)The transpose of a symmetri...
Explanation:
a) The transpose of a symmetric matrix need not be a symmetric matrix:
A square matrix is called symmetric if it is equal to its transpose. In other words, if A is a symmetric matrix, then A = A^T. However, the transpose of a symmetric matrix need not be symmetric. This is because the transpose operation simply interchanges the rows and columns of a matrix, without any change in the elements. Therefore, the property of symmetry is not preserved under the transpose operation.
b) If A and B are symmetric matrices of the same order, then AB and BA must be symmetric matrices:
If A and B are symmetric matrices of the same order, then the product AB need not be symmetric. However, the product BA is always symmetric. This can be proven by using the properties of matrix multiplication and the fact that A and B are symmetric matrices.
Let's consider AB first. The product of two matrices is obtained by multiplying the rows of the first matrix with the corresponding columns of the second matrix. In general, the order of multiplication matters in matrix multiplication. Therefore, AB need not be symmetric.
Now, let's consider BA. Since A and B are symmetric matrices, we have A = A^T and B = B^T. Therefore, BA = (B^T)(A^T) = (AB)^T = (AB). So, BA is equal to its transpose, which means BA is symmetric.
c) If A is a symmetric matrix, then all positive integral powers of A are symmetric matrices:
If A is a symmetric matrix, then A = A^T. To calculate the powers of A, we need to multiply A by itself. Since A is symmetric, we have A^2 = A * A = (A^T)(A) = (A)(A^T) = A^T * A = A^2. Therefore, all the positive integral powers of A are equal to A, which means they are also symmetric matrices.
d) If A is any square matrix, then A + A^T is always a symmetric matrix:
If A is any square matrix, then the sum of A and its transpose is always a symmetric matrix. This can be proven by considering the definition of symmetry and the properties of matrix addition and transpose operation. Let's denote B = A + A^T.
To show that B is symmetric, we need to prove that B = B^T. Using the definition of B, we have B^T = (A + A^T)^T = (A^T)^T + (A^T)^T = A + A^T = B. Therefore, B is equal to its transpose, which means B is symmetric.
Hence, the correct answers are b) If A and B are symmetric matrices of the same order, then AB and BA must be symmetric matrices, c) If A is a symmetric matrix, then all positive integral powers of A are symmetric matrices, and d) If A is any square matrix, then A + A^T is always a symmetric matrix.
a) The transpose of a symmetric matrix need not be a symmetric matrix:
A square matrix is called symmetric if it is equal to its transpose. In other words, if A is a symmetric matrix, then A = A^T. However, the transpose of a symmetric matrix need not be symmetric. This is because the transpose operation simply interchanges the rows and columns of a matrix, without any change in the elements. Therefore, the property of symmetry is not preserved under the transpose operation.
b) If A and B are symmetric matrices of the same order, then AB and BA must be symmetric matrices:
If A and B are symmetric matrices of the same order, then the product AB need not be symmetric. However, the product BA is always symmetric. This can be proven by using the properties of matrix multiplication and the fact that A and B are symmetric matrices.
Let's consider AB first. The product of two matrices is obtained by multiplying the rows of the first matrix with the corresponding columns of the second matrix. In general, the order of multiplication matters in matrix multiplication. Therefore, AB need not be symmetric.
Now, let's consider BA. Since A and B are symmetric matrices, we have A = A^T and B = B^T. Therefore, BA = (B^T)(A^T) = (AB)^T = (AB). So, BA is equal to its transpose, which means BA is symmetric.
c) If A is a symmetric matrix, then all positive integral powers of A are symmetric matrices:
If A is a symmetric matrix, then A = A^T. To calculate the powers of A, we need to multiply A by itself. Since A is symmetric, we have A^2 = A * A = (A^T)(A) = (A)(A^T) = A^T * A = A^2. Therefore, all the positive integral powers of A are equal to A, which means they are also symmetric matrices.
d) If A is any square matrix, then A + A^T is always a symmetric matrix:
If A is any square matrix, then the sum of A and its transpose is always a symmetric matrix. This can be proven by considering the definition of symmetry and the properties of matrix addition and transpose operation. Let's denote B = A + A^T.
To show that B is symmetric, we need to prove that B = B^T. Using the definition of B, we have B^T = (A + A^T)^T = (A^T)^T + (A^T)^T = A + A^T = B. Therefore, B is equal to its transpose, which means B is symmetric.
Hence, the correct answers are b) If A and B are symmetric matrices of the same order, then AB and BA must be symmetric matrices, c) If A is a symmetric matrix, then all positive integral powers of A are symmetric matrices, and d) If A is any square matrix, then A + A^T is always a symmetric matrix.
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Which of the following(s) is/are correct?a)The transpose of a symmetric matrix need not be summetric matrix.b)If A and B are symmetric matrix of same order, then AB + BA must be symmetric matrix.c)If A is symmetric matrix, then all positive integral powers of A are symmetric matrices.d)If A is any square matrix, then A + A'is always symmetricCorrect answer is option 'B,C,D'. Can you explain this answer?
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